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Continuous Random Variables Lecture 22 Section 7.5.4 Mon, Feb 25, 2008.

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1 Continuous Random Variables Lecture 22 Section 7.5.4 Mon, Feb 25, 2008

2 Random Variables Random variable Discrete random variable Continuous random variable

3 Continuous Probability Distribution Functions Continuous Probability Distribution Function (pdf) – For a random variable X, it is a function with the property that the area below the graph of the function between any two points a and b equals the probability that a ≤ X ≤ b. Remember, AREA = PROPORTION = PROBABILITY

4 Example The TI-83 will return a random number between 0 and 1 if we enter rand and press ENTER. These numbers have a uniform distribution from 0 to 1. Let X be the random number returned by the TI-83.

5 Example The graph of the pdf of X. x f(x)f(x) 01 1

6 Example What is the probability that the random number is at least 0.3?

7 Example What is the probability that the random number is at least 0.3? x f(x)f(x) 01 1 0.3

8 Example What is the probability that the random number is at least 0.3? x f(x)f(x) 01 1 0.3

9 Example What is the probability that the random number is at least 0.3? x f(x)f(x) 01 1 0.3

10 Area = 0.7 Example Probability = 70%. x f(x)f(x) 01 1 0.3

11 0.25 Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x) 01 1 0.75

12 Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x) 01 1 0.250.75

13 Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x) 01 1 0.250.75

14 Example Probability = 50%. x f(x)f(x) 01 1 0.250.75 Area = 0.5

15 Uniform Distributions The uniform distribution from a to b is denoted U(a, b). ab 1/(b – a) x

16 A Non-Uniform Distribution Consider this distribution. 510 x

17 A Non-Uniform Distribution What is the height? 510 ? x

18 A Non-Uniform Distribution The height is 0.4. 510 0.4 x

19 A Non-Uniform Distribution What is the probability that 6  X  8? 510 0.4 x 68

20 A Non-Uniform Distribution It is the same as the area between 6 and 8. 510 0.4 x 68

21 Uniform Distributions The uniform distribution from a to b is denoted U(a, b). ab 1/(b – a)

22 Hypothesis Testing (n = 1) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). One value of X is sampled (n = 1).

23 Hypothesis Testing (n = 1) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). One value of X is sampled (n = 1). If X is more than 0.75, then H 0 will be rejected.

24 Hypothesis Testing (n = 1) Distribution of X under H 0 : Distribution of X under H 1 : 00.511.5 1 00.511.5 1

25 Hypothesis Testing (n = 1) What are  and  ? 00.511.5 1 00.511.5 1

26 Hypothesis Testing (n = 1) What are  and  ? 0.75 00.511.5 1 00.511.5 1

27 Hypothesis Testing (n = 1) What are  and  ? 0.75 00.511.5 1 00.511.5 1 Acceptance RegionRejection Region

28 Hypothesis Testing (n = 1) What are  and  ? 0.75 00.511.5 1 00.511.5 1

29 Hypothesis Testing (n = 1) What are  and  ? 0.75 00.511.5 1 00.511.5 1   = ¼ = 0.25

30 Hypothesis Testing (n = 1) What are  and  ? 0.75 00.511.5 1 00.511.5 1   = ¼ = 0.25   = ¼ = 0.25 0.75

31 Example Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together. Let X 2 = the average of the two random numbers. What is the pdf of X 2 ?

32 Example The graph of the pdf of X 2. y f(y)f(y) 00.51 ?

33 Example The graph of the pdf of X 2. y f(y)f(y) 00.51 2 Area = 1

34 Example What is the probability that X 2 is between 0.25 and 0.75? y f(y)f(y) 00.510.250.75 2

35 Example What is the probability that X 2 is between 0.25 and 0.75? y f(y)f(y) 00.510.250.75 2

36 Example The probability equals the area under the graph from 0.25 to 0.75. y f(y)f(y) 00.5 2 10.250.75

37 Example Cut it into two simple shapes, with areas 0.25 and 0.5. y f(y)f(y) 00.510.250.75 2 Area = 0.5 Area = 0.25 0.5

38 Example The total area is 0.75. y f(y)f(y) 00.510.250.75 2 Area = 0.75

39 Hypothesis Testing (n = 2) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). Two values of X are sampled (n = 2). Let X 2 be the average. If X 2 is more than 0.75, then H 0 will be rejected.

40 Hypothesis Testing (n = 2) Distribution of X 2 under H 0 : Distribution of X 2 under H 1 : 00.511.5 2 00.511.5 2

41 Hypothesis Testing (n = 2) What are  and  ? 00.511.5 2 00.511.5 2

42 Hypothesis Testing (n = 2) What are  and  ? 00.511.5 2 00.511.5 2 0.75

43 Hypothesis Testing (n = 2) What are  and  ? 00.511.5 2 00.511.5 2 0.75

44 Hypothesis Testing (n = 2) What are  and  ? 00.511.5 2 00.511.5 2 0.75   = 1/8 = 0.125

45 Hypothesis Testing (n = 2) What are  and  ? 00.511.5 2 00.511.5 2 0.75   = 1/8 = 0.125   = 1/8 = 0.125

46 Conclusion By increasing the sample size, we can lower both  and  simultaneously.

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